The oversimplified version would be that they're exploiting time dilation to store light from a laser and release it across a shorter interval, like charging and discharging a battery or capacitor.

Made up numbers for illustration - the laser pulses they're actually talking about take place in quadrillionths of a second.

Take a laser and fire it at a mirror for 2 seconds. Normally, the reflection from that surface would last as long as the laser is firing - for the whole 2 seconds. The reflection will always be weaker than the laser because basic physics, can't have more than 100% efficiency.

The mirror in the article though is a wall of plasma that moves so quickly that it's subject to noticeable time dilation. Now, instead of the reflection lasting the 2 seconds of our laser pulse, it's being 'compressed' by time dilation into 1 second.

As long as the compression factor is greater than the fraction of the energy lost to the mirror and refocussing the reflection, this is effectively an increase in peak output power but at the cost of reduced overall output power (as your output power for part of the pulse is nil, and you have additional losses).